The sum of two numbers is 60 and their product is 875 . The HCF of the numbers is

Options

A. 6
B. 8
C. 5
D. 7

Answer

To solve the problem, we need to find two numbers whose sum is 60 and product is 875, and then determine their Highest Common Factor (HCF).

### Step-by-Step Solution:

1. **Define the Variables:**
Let the two numbers be \( x \) and \( y \).
According to the problem, we have:
\[ x + y = 60 \quad (1) \]
\[ xy = 875 \quad (2) \]

2. **Express \( y \) in terms of \( x \):**
From equation (1), we can express \( y \) as:
\[ y = 60 - x \quad (3) \]

3. **Substitute \( y \) in equation (2):**
Substitute equation (3) into equation (2):
\[ x(60 - x) = 875 \]
This simplifies to:
\[ 60x - x^2 = 875 \]
Rearranging gives:
\[ x^2 - 60x + 875 = 0 \quad (4) \]

4. **Solve the Quadratic Equation:**
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -60, c = 875 \):
\[ b^2 - 4ac = (-60)^2 - 4 \cdot 1 \cdot 875 = 3600 - 3500 = 100 \]
Now substituting back into the formula:
\[ x = \frac{60 \pm \sqrt{100}}{2} = \frac{60 \pm 10}{2} \]
This gives us two possible values for \( x \):
\[ x = \frac{70}{2} = 35 \quad \text{or} \quad x = \frac{50}{2} = 25 \]

5. **Find the Corresponding \( y \) Values:**
Using equation (3):
- If \( x = 35 \), then \( y = 60 - 35 = 25 \).
- If \( x = 25 \), then \( y = 60 - 25 = 35 \).

Thus, the two numbers are \( 35 \) and \( 25 \).

6. **Calculate the HCF:**
To find the HCF of \( 35 \) and \( 25 \):
- The prime factorization of \( 35 = 5 \times 7 \).
- The prime factorization of \( 25 = 5 \times 5 \).
- The common factor is \( 5 \).

Thus, the HCF of the two numbers is \( 5 \).

### Final Answer:
The HCF of the two numbers is \( 5 \).

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